The paper presents the diagrammatic calculus CL, which combines features of tree, Euler-type, Venn-type diagrams and squares of opposition. In its basic form, `CL' (= Cubus Logicus) organizes terms in the form of a square or cube. By applying the arrows of the square of opposition to CL, judgments and inferences can be displayed. Thus CL offers on the one hand an intuitive method to display ontologies and on the other hand a diagrammatic tool to check inferences. The paper focuses (...) mainly on the adaptation of the square of opposition in CL and offers an algebraic notation, which corresponds to the diagrammatic representation. (shrink)

The Aristotelian square of oppositions is a well-known diagram in logic and linguistics. In recent years, several extensions of the square have been discovered. However, these extensions have failed to become as widely known as the square. In this paper we argue that there is indeed a fundamental difference between the square and its extensions, viz., a difference in informativity. To do this, we distinguish between concrete Aristotelian diagrams and, on a more abstract level, the Aristotelian geometry. We then (...) introduce two new logical geometries, and develop a formal, well-motivated account of their informativity. This enables us to show that the square is strictly more informative than many of the more complex diagrams. (shrink)

Whereas geometrical oppositions (logical squares and hexagons) have been so far investigated in many fields of modal logic (both abstract and applied), the oppositional geometrical side of “deontic logic” (the logic of “obligatory”, “forbidden”, “permitted”, . . .) has rather been neglected. Besides the classical “deontic square” (the deontic counterpart of Aristotle’s “logical square”), some interesting attempts have nevertheless been made to deepen the geometrical investigation of the deontic oppositions: Kalinowski (La logique des normes, PUF, Paris, 1972) has proposed (...) a “deontic hexagon” as being the geometrical representation of standard deontic logic, whereas Joerden (jointly with Hruschka, in Archiv für Rechtsund Sozialphilosophie 73:1, 1987), McNamara (Mind 105:419, 1996) and Wessels (Die gute Samariterin. Zur Struktur der Supererogation, Walter de Gruyter, Berlin, 2002) have proposed some new “deontic polygons” for dealing with conservative extensions of standard deontic logic internalising the concept of “supererogation”. Since 2004 a new formal science of the geometrical oppositions inside logic has appeared, that is “ n -opposition theory”, or “NOT”, which relies on the notion of “logical bi-simplex of dimension m ” ( m = n − 1). This theory has received a complete mathematical foundation in 2008, and since then several extensions. In this paper, by using it, we show that in standard deontic logic there are in fact many more oppositional deontic figures than Kalinowski’s unique “hexagon of norms” (more ones, and more complex ones, geometrically speaking: “deontic squares”, “deontic hexagons”, “deontic cubes”, . . ., “deontic tetraicosahedra”, . . .): the real geometry of the oppositions between deontic modalities is composed by the aforementioned structures (squares, hexagons, cubes, . . ., tetraicosahedra and hyper-tetraicosahedra), whose complete mathematical closure happens in fact to be a “deontic 5-dimensional hyper-tetraicosahedron” (an oppositional very regular solid). (shrink)

To appear in Jean-Yves Béziau (ed.) Proc. First World Congress on the Square of Opposition.

In 1995 Slater argued both against Priest’s paraconsistent system LP (1979) and against paraconsistency in general, invoking the fundamental opposition relations ruling the classical logical square. Around 2002 Béziau constructed a double defence of paraconsistency (logical and philosophical), relying, in its philosophical part, on Sesmat’s (1951) and Blanche’s (1953) “logical hexagon”, a geometrical, conservative extension of the logical square, and proposing a new (tridimensional) “solid of opposition”, meant to shed new light on the point raised by Slater. By using n-opposition (...) theory (NOT) we analyse Beziau’s anti-Slater move and show both its right intuitions and its technical limits. Moreover, we suggest that Slater’s criticism is much akin to a well-known one by Suszko (1975) against the conceivability of many-valued logics. This last criticism has been addressed by Malinowski (1990) and Shramko and Wansing (2005), who developed a family of tenable logical counter-examples to it: trans-Suszkian systems are radically many-valued. This family of new logics has some strange logical features, essentially: each system has more than one consequence operator. We show that a new, deeper part of the aforementioned geometry of logical oppositions (NOT), the “logical poly-simplexes of dimension m”, generates new logical-geometrical structures, essentially many-valued, which could be a very natural (and intuitive) geometrical counterpart to the “strange”, new, non-Suszkian logics of Malinowski, Shramko and Wansing. By a similar move, the geometry of opposition therefore sheds light both on the foundations of paraconsistent logics and on those of many-valued logics. (shrink)

The city has featured as a central image in utopian thought. In planning the foundation of the new and ideal city there is a close interconnection between ideas about urban form and the vision of the moral good. The spatial structure of the ideal city in these visions is a framing device that embodies and articulates not only political philosophy but is itself an articulation of moral and cosmological systems. This paper analyses three different utopian moments in three different historical (...) epochs – Tommaso Campanella’s City of the Sun (1602), the Choson dynasty foundation of the city of Seoul (1395) and the modernist utopian urbanism of the controversial Le Corbusier (1887–1965). In this analysis attention is drawn to the cosmological and moral visions articulated in these three ‘images of the city’ (Lynch). The opposition between rationalistic/mechanistic and religious/traditional urban design can prove to be an oversimplification that obscures the complex interrelations between the moral geometry and the natural alignments of the ideal urban form. (shrink)

An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. Io finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers.

Since antiquity, opposed concepts such as the One and the Many, the Finite and the Infinite, and the Absolute and the Relative, have been a driving force in philosophical, scientific, and mathematical thought. Yet they have also given rise to perplexing problems and conceptual paradoxes which continue to haunt scientists and philosophers. In _Oppositions and Paradoxes_, John L. Bell explains and investigates the paradoxes and puzzles that arise out of conceptual oppositions in physics and mathematics. In the process, Bell not (...) only motivates abstract conceptual thinking about the paradoxes at issue, but he also offers a compelling introduction to central ideas in such otherwise-difficult topics as non-Euclidean geometry, relativity, and quantum physics. These paradoxes are often as fun as they are flabbergasting. Consider, for example, the famous Tristram Shandy paradox: an immortal man composing an autobiography so slowly as to require a year of writing to describe each day of his life — he would, if he had infinite time, presumably never complete the work, although no individual part of it would remain unwritten. Or think of an office mailbox labelled “mail for those with no mailbox”—if this is a person’s mailbox, how can they possibly have “no mailbox”? These and many other paradoxes straddle the boundary between physics and metaphysics, and demonstrate the hidden difficulty in many of our most basic concepts. (shrink)

The analysis of the golden Mithras’ bas-relief in the Museum of the Baths of Diocletian in Rome confirms the Platonic Chiasma. The scene admits two diagonals starting from each corner. One passes through the sun and the other through the moon. The sun god is also shown with an object in his left hand, which may be a soul or a sacred heart. This would confirm that the slab shows the opposition between metempsychosis (lunar) and resurrection (solar). The analysis of (...) the Barberini fresco is of great interest. It shows that the Platonic Chiasma falls exactly within the celestial gates of Capricorn and Cancer. The solar line is superimposed on a line already present on the fresco. Finally, sacred geometry helps to explain some strange-looking reliefs from the Danube region. (shrink)

Some Carrollian posthumous manuscripts reveal, in addition to his famous ‘logical diagrams’, two mysterious ‘logical charts’. The first chart, a strange network making out of fourteen logical sentences a large 2D ‘triangle’ containing three smaller ones, has been shown equivalent—modulo the rediscovery of a fourth smaller triangle implicit in Carroll's global picture—to a 3D tetrahedron, the four triangular faces of which are the 3+1 Carrollian complex triangles. As it happens, such an until now very mysterious 3D logical shape—slightly deformed—has been (...) rediscovered, independently from Carroll and much later, by a logician , a mathematician and a linguist studying the geometry of the ‘opposition relations’, that is, the mathematical generalisations of the ‘logical square’. We show that inside what is called equivalently ‘n-opposition theory’, ‘oppositional geometry’ or ‘logical geometry’, Carroll's first chart corresponds exactly, duly reshap.. (shrink)

In order to elucidate his logical analysis of modal quantified propositions (e.g. ‘all men are necessarily mortal’), the 14th century philosopher John Buridan constructed a modal octagon of oppositions. In the present paper we study this modal octagon from the perspective of contemporary logical geometry. We argue that the modal octagon contains precisely six squares of opposition as subdiagrams, and classify these squares based on their logical properties. On a more abstract level, we show that Buridan’s modal octagon precisely (...) captures the interaction between two classical squares of opposition, viz. one for the quantifiers and one for the modalities. Finally, we argue that several aspects of our contemporary formal analyses were already hinted at by Buridan himself. (shrink)

Brown and Pooley’s ‘dynamical approach’ to physical theories asserts, in opposition to the orthodox position on physical geometry, that facts about physical geometry are grounded in, or explained by, facts about dynamical fields, not the other way round. John Norton has claimed that the proponent of the dynamical approach is illicitly committed to spatiotemporal presumptions in ‘constructing’ space-time from facts about dynamical symmetries. In this article, I present an abstract, algebraic formulation of field theories and demonstrate that the (...) proponent of the dynamical approach is not committed, in special relativity, to the illicit presumptions to which Norton refers. (shrink)

The square of opposition is a diagram related to a theory of oppositions that goes back to Aristotle. Both the diagram and the theory have been discussed throughout the history of logic. Initially, the diagram was employed to present the Aristotelian theory of quantification, but extensions and criticisms of this theory have resulted in various other diagrams. The strength of the theory is that it is at the same time fairly simple and quite rich. The theory of oppositions has recently (...) become a topic of intense interest due to the development of a general geometry of opposition (polygons and polyhedra) with many applications. A congress on the square with an interdisciplinary character has been organized on a regular basis (Montreux 2007, Corsica 2010, Beirut 2012, Vatican 2014, Rapa Nui 2016). The volume at hand is a sequel to two successful books: The Square of Opposition - A General Framework of Cognition, ed. by J.-Y. Béziau & G. Payette, as well as Around and beyond the Square of Opposition, ed. by J.-Y. Béziau & D. Jacquette, and, like those, a collection of selected peer-reviewed papers. The idea of this new volume is to maintain a good equilibrium between history, technical developments and applications. The volume is likely to attract a wide spectrum of readers, mathematicians, philosophers, linguists, psychologists and computer scientists, who may range from undergraduate students to advanced researchers. (shrink)

The Cartan equations defining simple spinors (renamed “pure” by C. Chevalley) are interpreted as equations of motion in compact momentum spaces, in a constructive approach in which at each step the dimensions of spinor space are doubled while those of momentum space increased by two. The construction is possible only in the frame of the geometry of simple or pure spinors, which imposes contraint equations on spinors with more than four components, and then momentum spaces result compact, isomorphic to (...) invariant-mass-spheres imbedded in each other, since the signatures result steadily Lorentzian; starting from dimension four (Minkowski) up to dimension ten with Clifford algebra ℂℓ(1, 9), where the construction naturally ends. The equations of motion met in the construction are most of those traditionally postulated ad hoc: from Weyl equations for neutrinos (and Maxwell's) to Majorana ones, to those for the electroweak model and for the nucleons interacting with the pseudoscalar pion, up to those for the 3 baryon-lepton families, steadily progressing from the description of lower energy phenomena to that of higher ones. The 3 division algebras: complex numbers, quaternions and octonions appear to be strictly correlated with Clifford algebras and then with this spinor-geometrical approach, from which they appear to gradually emerge in the construction, where they play a basic role for the physical interpretation: at the third step complex numbers generate U(1), possible origin of the electric charge and of the existence of charged—neutral fermion pairs, explaining also easily the opposite charges of proton-electron. Another U(1) appears to generate the strong charge at the fourth step. Quaternions generate the signature of space-time at the first step, the SU(2) internal symmetry of isospin and, in the gauge term, the SU(2) L one, of the electroweak model at the third step; they are also at the origin of 3 families; in number equal to that of quaternion imaginary units. At the fifth and last step octonions generate the SU(3) internal symmetry of flavour, with SU(2) isospin subgroup and, in the gauge term, the one of color, correlated with SU(2) L of the electroweak model. These 3 division algebras seem then to be at the origin of charges, families and of the groups of the Standard model. In this approach there seems to be no need of higher dimensional (>4) space-time, here generated by the four Poincaré translations, and dimensional reduction from ℂℓ(1,9) to ℂℓ(1,3) is equivalent to decoupling of the equations of motion. This spinor-geometrical approach is compatible with that based on strings, since these may be expressed bilinearly (as integrals) in terms of Majorana–Weyl simple or pure spinors which are admitted by ℂℓ(1, 9) = R(32). (shrink)

La philosophie de la géométrie de Hölder, si l’on s’en tient à une lecture superficielle, est la part la plus problématique de son épistémologie. Il soutient que la géométrie est fondée sur l’expérience à la manière de Helmholtz, malgré les objections sérieuses de Poincaré. Néanmoins, je pense que la position de Hölder mérite d’être discutée pour deux motifs. Premièrement, ses implications méthodologiques furent importantes pour le développement de son épistémologie. Deuxièmement, Poincaré utilise l’opposition entre le kantisme et l’empirisme comme un (...) argument pour justifier son conventionnalisme géométrique. Cependant, Hölder montre qu’une stratégie alternative n’est pas exclue: il sait tirer parti des objections kantiennes pour développer un empirisme cohérent. En même temps, surtout dans Die mathematische Methode [Holder 1924], il adopte aussi bien les expressions que les conceptions de Kant. Dans mon article, je considère d’abord les arguments de Hölder pour la méthode déductive en géométrie dans Anschauung und Denken in der Geometrie [Holder 1900], en relation avec sa façon d’aborder la théorie de la quantité [Holder 1901]. Ensuite, j’examine son rapport avec Kant. À mon sens, les considérations méthodologiques de Hölder lui permettent de préfigurer une relativisation de l’a priori. (shrink)

La philosophie de la géométrie de Hölder, si l’on s’en tient à une lecture superficielle, est la part la plus problématique de son épistémologie. Il soutient que la géométrie est fondée sur l’expérience à la manière de Helmholtz, malgré les objections sérieuses de Poincaré. Néanmoins, je pense que la position de Hölder mérite d’être discutée pour deux motifs. Premièrement, ses implications méthodologiques furent importantes pour le développement de son épistémologie. Deuxièmement, Poincaré utilise l’opposition entre le kantisme et l’empirisme comme un (...) argument pour justifier son conventionnalisme géométrique. Cependant, Hölder montre qu’une stratégie alternative n’est pas exclue: il sait tirer parti des objections kantiennes pour développer un empirisme cohérent. En même temps, surtout dans Die mathematische Methode [Holder 1924], il adopte aussi bien les expressions que les conceptions de Kant. Dans mon article, je considère d’abord les arguments de Hölder pour la méthode déductive en géométrie dans Anschauung und Denken in der Geometrie [Holder 1900], en relation avec sa façon d’aborder la théorie de la quantité [Holder 1901]. Ensuite, j’examine son rapport avec Kant. À mon sens, les considérations méthodologiques de Hölder lui permettent de préfigurer une relativisation de l’a priori. (shrink)

This paper explores the interplay between logic-sensitivity and bitstring semantics in the square of opposition. Bitstring semantics is a combinatorial technique for representing the formulas that appear in a logical diagram, while logic-sensitivity entails that such a diagram may depend, not only on the formulas involved, but also on the logic with respect to which they are interpreted. These two topics have already been studied extensively in logical geometry, and are thus well-understood by themselves. However, the precise details of (...) their interplay turn out to be far more complicated. In particular, the paper describes an elegant and natural interaction between bitstrings and logic-sensitivity, which makes perfect sense when bitstrings are viewed as purely combinatorial entities. However, when we view bitstrings as semantically meaningful entities (which is actually the standard perspective, cf. the term ‘bitstring semantics’!), this interaction does not seem to have a full and equally natural counterpart. The paper describes some attempts to address this situation, but all of them are ultimately found wanting. For now, it thus remains an open problem to capture this interaction between bitstrings and logic-sensitivity from a semantic (rather than merely a combinatorial) perspective. (shrink)

This paper studies John Buridan's octagons of opposition for the de re modal propositions and the propositions of unusual construction. Both Buridan himself and the secondary literature have emphasized the strong similarities between these two octagons (as well as a third one, for propositions with oblique terms). In this paper, I argue that the interconnection between both octagons is more subtle than has previously been thought: if we move beyond the Aristotelian relations, and also take Boolean considerations into account, then (...) the strong analogy between Buridan's octagons starts to break down. These differences in Boolean structure can already be discerned within the octagons themselves; on a more abstract level, they lead to these two octagons having different degrees of Boolean complexity (i.e. Boolean closures of different sizes). These results are obtained by means of bitstring analysis, which is one of the key tools from contemporary logical geometry. Finally, I argue that this historical investigation is directly relevant for the theoretical framework of logical geometry, and discuss how it helps us to address certain open questions in this framework. (shrink)

La philosophie de la géométrie de Hölder, si l’on s’en tient à une lecture superficielle, est la part la plus problématique de son épistémologie. Il soutient que la géométrie est fondée sur l’expérience à la manière de Helmholtz, malgré les objections sérieuses de Poincaré. Néanmoins, je pense que la position de Hölder mérite d’être discutée pour deux motifs. Premièrement, ses implications méthodologiques furent importantes pour le développement de son épistémologie. Deuxièmement, Poincaré utilise l’opposition entre le kantisme et l’empirisme comme un (...) argument pour justifier son conventionnalisme géométrique. Cependant, Hölder montre qu’une stratégie alternative n’est pas exclue: il sait tirer parti des objections kantiennes pour développer un empirisme cohérent. En même temps, surtout dans Die mathematische Methode [Holder 1924], il adopte aussi bien les expressions que les conceptions de Kant. Dans mon article, je considère d’abord les arguments de Hölder pour la méthode déductive en géométrie dans Anschauung und Denken in der Geometrie [Holder 1900], en relation avec sa façon d’aborder la théorie de la quantité [Holder 1901]. Ensuite, j’examine son rapport avec Kant. À mon sens, les considérations méthodologiques de Hölder lui permettent de préfigurer une relativisation de l’a priori. (shrink)

This paper examines the interplay of changelessness and change, being and becoming, from an historical and dialectical standpoint. Topological paradox is employed to elucidate the dynamic interweaving of these ontological opposites. The essay concludes by exploring the relevance of the dialectic to the question of human freedom.

The logical hexagon (or hexagon of opposition) is a strange, yet beautiful, highly symmetrical mathematical figure, mysteriously intertwining fundamental logical and geometrical features. It was discovered more or less at the same time (i.e. around 1950), independently, by a few scholars. It is the successor of an equally strange (but mathematically less impressive) structure, the “logical square” (or “square of opposition”), of which it is a much more general and powerful “relative”. The discovery of the former did not raise interest, (...) neither among logicians, nor among philosophers of logic, whereas the latter played a very important theoretical role (both for logic and philosophy) for nearly two thousand years, before falling in disgrace in the first half of the twentieth century: it was, so to say, “sentenced to death” by the so-called analytical philosophers and logicians. Contrary to this, since 2004 a new, unexpected promising branch of mathematics (dealing with “oppositions”) has appeared, “oppositional geometry” (also called “n-opposition theory”, “NOT”), inside which the logical hexagon (as well as its predecessor, the logical square) is only one term of an infinite series of “logical bi-simplexes of dimension m”, itself just one term of the more general infinite series (of series) of the “logical poly-simplexes of dimension m”. In this paper we recall the main historical and the main theoretical elements of these neglected recent discoveries. After proposing some new results, among which the notion of “hybrid logical hexagon”, we show which strong reasons, inside oppositional geometry, make understand that the logical hexagon is in fact a very important and profound mathematical structure, destined to many future fruitful developments and probably bearer of a major epistemological paradigm change. (shrink)

Logical geometry systematically studies Aristotelian diagrams, such as the classical square of oppositions and its extensions. These investigations rely heavily on the use of bitstrings, which are compact combinatorial representations of formulas that allow us to quickly determine their Aristotelian relations. However, because of their general nature, bitstrings can be applied to a wide variety of topics in philosophical logic beyond those of logical geometry. Hence, the main aim of this paper is to present a systematic technique for (...) assigning bitstrings to arbitrary finite fragments of formulas in arbitrary logical systems, and to study the logical and combinatorial properties of this technique. It is based on the partition of logical space that is induced by a given fragment, and sheds new light on a number of interesting issues, such as the logic-dependence of the Aristotelian relations and the subtle interplay between the Aristotelian and Boolean structure of logical fragments. Finally, the bitstring technique also allows us to systematically analyze fragments from contemporary logical systems, such as public announcement logic, which could not be done before. (shrink)

In recent years, a number of authors have started studying Aristotelian diagrams containing metalogical notions, such as tautology, contradiction, satisfiability, contingency, strong and weak interpretations of contrariety, etc. The present paper is a contribution to this line of research, and its main aims are both to extend and to deepen our understanding of metalogical diagrams. As for extensions, we not only study several metalogical decorations of larger and less widely known Aristotelian diagrams, but also consider metalogical decorations of another type (...) of logical diagrams, viz. duality diagrams. At a more fundamental level, we present a unifying perspective which sheds new light on the connections between new and existing metalogical diagrams, as well as between object- and metalogical diagrams. Overall, the paper studies two types of logical diagrams and four kinds of metalogical decorations. (shrink)

It has been recently argued that the well-known square of opposition is a gathering that can be reduced to a one-dimensional figure, an ordered line segment of positive and negative integers [3]. However, one-dimensionality leads to some difficulties once the structure of opposed terms extends to more complex sets. An alternative algebraic semantics is proposed to solve the problem of dimensionality in a systematic way, namely: partition (or bitstring) semantics. Finally, an alternative geometry yields a new and unique pattern (...) of oppositions that proceeds with colored diagrams and an increasing set of bitstrings. (shrink)

The aim of this paper is to point out significant and meaningful overlapping between several styles of scientific thinking, as they were proposed by Crombie (1981) and discussed by Hacking (1985; 2009). This paper is divided in four sections. First, I examine an interpretation made by Barnes (2004) about the incompatibility among scientific styles. As explained by its author, this interpretation denies any possibility of similarities between styles of scientific reasoning. In opposition, the following sections of this paper include explanations (...) of relevant characteristics of Geometry, as stated in Euclid's Elements, which are also present in other three scientific styles: modeling, experimental exploration and taxonomic style (and some of the shared characteristics are even used to define such styles). By stressing out these characteristics I argue that these four styles discussed by Hacking are not so different from each other, in fact, they overlap and may even be abridged into one foundational style: geometric. El objetivo de este artículo consiste en explicar traslapes importantes y significativos entre varios estilos de pensamiento científico, tal como fueron propuestos por Crombie (1981) y discutidos por Hacking (1985; 2009). El presente artículo está dividido en cuatro secciones. Primero, se examina la interpretación de Barnes (2004) sobre la incompatibilidad entre diversos estilos. Según este autor, no existe posibilidad alguna de encontrar similitudes en los estilos de razonamiento científico. En oposición a esta interpretación, las siguientes secciones explican características relevantes de la Geometría, tal como fueron plasmadas en Los Elementos de Euclides, y que están presentes en otros tres estilos: de modelación, de experimentación y taxonómico (las características compartidas incluso definen algunos de estos estilos). Destacar estas características me permite argumentar que estos cuatro estilos discutidos por Hacking no son tan diferentes entre sí, de hecho, se traslapan e incluso pueden ser condensados en un estilo fundacional: el geométrico. (shrink)

We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model S as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In S , all definable sets of reals are (...) Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace −∫ (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy. (shrink)

Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are (...) devoted. At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)

The present first part about the eventual completeness of mathematics (called “Hilbert mathematics”) is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. (...) Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering “theorem”, transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for “Hilbert mathematics” accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel’s papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part. (shrink)

The hexagonal structure for ‘the geometry of logical opposition’, as coming from Aristoteles–Apuleius square and Sesmat–Blanché hexagon, is presented here in connection with, on the one hand, geometrical ideas on duality on triangles (construction of ‘companion’), and on the other hand, constructions of tripartitions, emphasizing that these are exactly cases of borromean objects. Then a new case of a logical interest introduced here is the double magic tripartition determining the semi-ring ${\mathcal{B}_3}$ and this is a borromean object again, in (...) the heart of the semi-ring ${{\rm Mat}_{3}(\mathbb{B}_{\rm Alg})}$ . With this example we understand better in which sense the borromean object is a deepening of the hexagon, in a logical vein. Then, and this is our main objective here, the Post-Mal’cev full iterative algebra ${\mathbb{P}_4 = \mathbb{P}(\mathbb{F}_4)}$ of functions of all arities on ${\mathbb{F}_4}$ , is proved to be a borromean object, generated by three copies of ${\mathbb{P}_2}$ in it. This fact is induced by a hexagonal structure of the field ${\mathbb{F}_4}$ . This hexagonal structure is seen as precisely a geometrical addition to standard boolean logic, exhibiting ${\mathbb{F}_4}$ as a ‘boolean manifold’. This structure allows to analyze also ${\mathbb{P}_4}$ as generated by adding to a boolean set of logical functions a very special modality, namely the Frobenius squaring map in ${\mathbb{F}_4}$ . It is related to the splitting of paradoxes, to modified logic, to specular logic. It is a setting for a theory of paradoxical sentences, seen as computations of movements on the bi-hexagonal link among the 12 classical logics on a set of 4 values. (shrink)

In logical geometry, Aristotelian diagrams are studied in a precise and systematic way. Although there has recently been a good amount of progress in logical geometry, it is still unknown which underlying mathematical framework is best suited for formalizing the study of these diagrams. Hence, in this paper, the main aim is to formulate such a framework, using the powerful language of category theory. We build multiple categories, which all have Aristotelian diagrams as their objects, while having different (...) kinds of morphisms between these diagrams. The categories developed here are assessed according to their ability to generalize previous work from logical geometry as well as their interesting category-theoretical properties. According to these evaluations, the most promising category has as its morphisms those functions on fragments that increase in informativity on both the opposition and implication relations. Focusing on this category can significantly increase the effectiveness of further research in logical geometry. (shrink)

RésuméCet article examine l'opposition entre preuve directe et preuve per impossibile, introduite en géométrie par al-Sijzī dans un mémoire intitulé Le côté n'est pas commensurable à la diagonale. À partir de cet exemple qu'il donne pour illustrer cette opposition, al-Sijzī défend en effet la supériorité de la preuve directe. L'article propose également l'editio princeps et la première traduction de ce mémoire.

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. (...) Starting with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)

In this first modern, critical assessment of the place of mathematics in Berkeley's philosophy and Berkeley's place in the history of mathematics, Douglas M. Jesseph provides a bold reinterpretation of Berkeley's work. Jesseph challenges the prevailing view that Berkeley's mathematical writings are peripheral to his philosophy and argues that mathematics is in fact central to his thought, developing out of his critique of abstraction. Jesseph's argument situates Berkeley's ideas within the larger historical and intellectual context of the Scientific Revolution. Jesseph (...) begins with Berkeley's radical opposition to the received view of mathematics in the philosophy of the late seventeenth and early eighteenth centuries, when mathematics was considered a "science of abstractions." Since this view seriously conflicted with Berkeley's critique of abstract ideas, Jesseph contends that he was forced to come up with a nonabstract philosophy of mathematics. Jesseph examines Berkeley's unique treatments of geometry and arithmetic and his famous critique of the calculus in The Analyst. By putting Berkeley's mathematical writings in the perspective of his larger philosophical project and examining their impact on eighteenth-century British mathematics, Jesseph makes a major contribution to philosophy and to the history and philosophy of science. (shrink)

This volume brings together those papers of mine which may be of interest not only to various specialists but also to philosophers. Many of my writings in mathematics were motivated by epistemological considerations; some papers originated in the critique of certain views that at one time dominated the discussions of the Vienna Cirele; others grew out of problems in teaching fundamental ideas of mathematics; sti II others were occasioned by personal relations with economists. Hence a wide range of subjects will (...) be discussed: epistemology, logic, basic concepts of pure and applied mathematics, philosophical ideas resulting from geometric studies, mathematical didactics and, finally, economics. The papers also span a period of more than fifty years. What unifies the various parts of the book is the spirit of searching for the elarification of basic concepts and methods and of articulating hidden ideas and tacit procedures. Part 1 ineludes papers published about 1930 which expound an idea that Carnap, after a short period of opposition in the Cirele, fully adopted ; and, under the name "Princip/e of To/erance", he eloquently formulated it in great generality in his book, Logica/ Syntax of Language, through which it was widely disseminated. "The New Logic" in Chapter 1 furthermore ineludes the first report to a larger public of Godel's epochal discovery presented among the great logic results of ali time. Chapter 2 is a translation of an often quoted 1930 paper presenting a detailed exposition and critique of intuitionism. (shrink)

Newtonian physics is based on Newtonian calculus applied to Newtonian dynamics. New paradigms such as ‘modified Newtonian dynamics’ change the dynamics, but do not alter the calculus. However, calculus is dependent on arithmetic, that is the ways we add and multiply numbers. For example, in special relativity we add and subtract velocities by means of addition β1⊕β2=tanh+tanh-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1\oplus \beta _2=\tanh \big +\tanh ^{-1}\big )$$\end{document}, although multiplication β1⊙β2=tanh·tanh-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} (...) \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1\odot \beta _2=\tanh \big \cdot \tanh ^{-1}\big )$$\end{document}, and division β1⊘β2=tanh/tanh-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta _1\oslash \beta _2=\tanh \big /\tanh ^{-1}\big )$$\end{document} do not seem to appear in the literature. The map fX=tanh-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{\mathbb{X}}=\tanh ^{-1}$$\end{document} defines an isomorphism of the arithmetic in X=\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}=$$\end{document} with the standard one in R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document}. The new arithmetic is projective and non-Diophantine in the sense of Burgin, 1977), while ultrarelativistic velocities are super-large in the sense of Kolmogorov, 1961). Velocity of light plays a role of non-Diophantine infinity. The new arithmetic allows us to define the corresponding derivative and integral, and thus a new calculus which is non-Newtonian in the sense of Grossman and Katz. Treating the above example as a paradigm, we ask what can be said about the set X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{X}}$$\end{document} of ‘real numbers’, and the isomorphism fX:X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{{\mathbb{X}}}:{\mathbb{X}}\rightarrow {\mathbb{R}}$$\end{document}, if we assume the standard form of Newtonian mechanics and general relativity but demand agreement with astrophysical observations. It turns out that the observable accelerated expansion of the Universe can be reconstructed with zero cosmological constant if fX≈0.8sinh/\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{\mathbb{X}}\approx 0.8\sinh /$$\end{document}. The resulting non-Newtonian model is exactly equivalent to the standard Newtonian one with ΩΛ=0.7\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _\Lambda =0.7$$\end{document}, ΩM=0.3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega _M=0.3$$\end{document}. Asymptotically flat rotation curves are obtained if ‘zero’, the neutral element 0X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0_{{\mathbb{X}}}$$\end{document} of addition, is nonzero from the point of view of the standard arithmetic of R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{R}}$$\end{document}. This implies fX-1=0X>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f^{-1}_{{\mathbb{X}}}=0_{{\mathbb{X}}}>0$$\end{document}. The opposition Diophantine versus non-Diophantine, or Newtonian versus non-Newtonian, is an arithmetic analogue of Euclidean versus non-Euclidean in geometry. We do not yet know if the proposed generalization ultimately removes any need of dark matter, but it will certainly change estimates of its parameters. Physics of the dark universe seems to be both geometry and arithmetic. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Kant on the Method of MathematicsEmily Carson1. INTRODUCTIONThis paper will touch on three very general but closely related questions about Kant’s philosophy. First, on the role of mathematics as a paradigm of knowledge in the development of Kant’s Critical philosophy; second, on the nature of Kant’s opposition to his Leibnizean predecessors and its role in the development of the Critical philosophy; and finally, on the specific role of intuition (...) in Kant’s philosophy of mathematics. One of the points that I want to make is that recognizing the importance of these first two issues is essential to giving an adequate account of the third. That is, only by appreciating how Kant uses the example of mathematical knowledge, and in particular the objections he had to previous accounts of such knowledge, can we understand the philosophical role that he ascribes to intuition in his own account of mathematics.I don’t propose to explain the details of Kant’s philosophy of mathematics, but rather to compare his views on mathematics in the pre-Critical Prize Essay of 1764 to those of the Critique of Pure Reason, with the aim of showing how Kant’s doctrine of construction in pure intuition arises out of a combination of two factors: first, his attempt to ground the distinction between the respective methods of mathematics and philosophy, and second, his concern to defend the reality of mathematical knowledge against the views of those he called ‘the metaphysicians.’ The first factor demands an account of the certainty of mathematics, the second requires an account of its content. I shall argue that there is a possible conflict between Kant’s accounts of these features in the Prize Essay, a possibility which is finally ruled out only by means of the Critical doctrine of pure intuition. So I hope to show that there is a significant philosophical role for intuition in Kant’s account of mathematical knowledge (over and above the more logical role attributed to it on readings [End Page 629] such as those of Friedman, Beth, and Hintikka1) in reconciling the content of mathematics and its certainty.2. THE MATHEMATICIANS VS. THE METAPHYSICIANSThroughout his early writings, Kant appealed to the evidence and certainty of mathematics; his attitude towards metaphysics, however, underwent a dramatic shift. In the Physical Monadology of 1756, Kant asked how metaphysics and geometry can be united,[f] or the former peremptorily denies that space is infinitely divisible, while the latter, with its usual certainty, asserts that it is infinitely divisible. Geometry contends that empty space is necessary for free motion, while metaphysics hisses the idea off the stage. Geometry holds universal attraction or gravitation to be hardly explicable by mechanical causes but shows that it derives from the forces which are inherent in bodies at rest and which act at a distance, whereas metaphysics dismisses the notion as an empty delusion of the imagination [1:475–6].2He attempted in this work to reconcile the respective positions of metaphysics and geometry regarding infinite divisibility, thereby demonstrating that “it is neither the case that the geometer is mistaken nor that the opinion to be found among metaphysicians deviates from the truth” [1:480]. But in a series of works from 1762–3, Kant reveals a markedly different attitude towards metaphysics. For example, in “The only possible argument in support of a demonstration of the existence of God,” Kant says:the mania for method and the imitation of the mathematician, who advances with a sure step along a well-surfaced road, have occasioned a large number of such mishaps on the slippery ground of metaphysics. These mishaps are constantly before one’s eyes, but there is little hope that people will be warned by them, or that they will learn to be more circumspect as a result [2:71].In contrast, Kant describes the geometer as uncovering “with the greatest certainty” the most secret properties of that which is extended [2:70]. But the less conciliatory attitude to metaphysics comes out most clearly in Kant’s actual procedure in “The only possible argument.” In the Seventh Reflection, he presents an attempt to explain the origin of... (shrink)

Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates illustrate the emerging idea (...) of mathematics as the science of structure. The article closes with some remarks on structuralist and nonstructuralist themes in Frege's own development of arithmetic. (shrink)

The central aim of this dissertation is to clarify, defend and develop a realist field ontology of the causal-inertial structure of spacetime forcefully advanced by Hermann Weyl. Weyl's field ontology of spacetime structure may roughly be described as follows. The Special and General as well as the non-relativistic spacetime theories are principle theories of spacetime structure. They all postulate various structural constraints, and events within spacetime are held to satisfy these constraints. When interpreted physically, these mathematical structures correspond to physical (...) structural fields on spacetime. ;The thesis begins with a discussion of the significance of Riemann's Dynamical Hypothesis within the context of the debate between geometrical realism and conventionalism. In particular, Grunbaum's unyielding insistence that Riemann's work must be seen as the immediate progenitor of his brand of geometrical conventionalism is challenged and shown to be unsupported by a textual analysis of the relevant works by Riemann. The first two chapters provide a historical/exegetical analysis of Riemann's Habilitationsvortrag. Chapter 3 presents a general account of geometrical conventionalism apart from issues of its alleged historical ancestry as seen by Grunbaum. This is followed by a critical examination of Grunbaum's interpretation of Riemann. ;Grunbaum has repeatedly argued that Weyl's interpretation of Riemann accords with his. It is shown, however, that Weyl's understanding of Riemann is the direct opposite of that of Grunbaum. Chapter 4 provides an examination of Weyl's philosophy of geometry and his interpretation of Riemann. It is shown that Weyl considers Riemann's work to be essentially progenitorial of his version of geometrical realism. . . . UMI. (shrink)

The book reconstructs the history of Western ethics. The approach chosen focuses the endless dialectic of moral codes, or different kinds of ethos, moral doctrines that are preached in order to bring about a reform of existing ethos, and ethical theories that have taken shape in the context of controversies about the ethos and moral doctrines as means of justifying or reforming moral doctrines. Such dialectic is what is meant here by the phrase ‘moral traditions’, taken as a name for (...) threads of moral discourse, made in turn of other interwoven threads, including transmission of shared codes, appeals to reform of prevailing custom, rational argument about the justification of some precept on the basis of some shared general teaching or principle, and rational argument about the ultimate basis for principles and justification of authoritative teaching. That is, the approach adopted to the reading of ethical texts depends on a firm belief in the fact that philosophers hardly created any ethical doctrine out of nothing. The main point this book tries to highlight is how different philosophical theories emerged and followed each other as a result of attempts at accounting for what was going on in the world of moral traditions. Changes were propelled by controversies between different schools, and highly abstract arguments were the unintended effects of moves made by controversialists forced to transform (and occasionally to turn upside down) their own doctrines in order to face the challenge posed by other arguments. This is the reason why the book examines not only texts that already enjoy pride of place in the history of philosophy (Aristotle, Kant, Hegel), but also other texts usually treated in the histories of religions (the Bible, the Talmud, the Quran), and others considered to be much less philosophical (Plutarchus, Pufendorf). -/- 1. Plato and a response to ethical scepticism. Two different traditions of morality in VI-V century Greece are described. The birth of philosophical questioning of traditional morality and temporal and spatial variation of custom is described within the context of the v century crisis, the demise of traditional aristocratic and tyrannical rule and the birth of democracy. Two conflicting answers to the challenge are reconstructed, namely conventionalist or immoralist theories formulated by the Sophists and the eudemonist and intellectualist Socratic theory. Plato’s own reformulation of Socrates answer to the Sophists is reconstructed. His psychological views, his classification of the four cardinal virtues and his political theory are described as parts of a unitary system, culminating in an extremely realist moral ontology identifying the idea of the good with the essence of the (moral and extra-moral) world itself. -/- 2. Aristotle and the invention of practical philosophy. Aristotle’s invention of practical philosophy as a field separated from first philosophy is shown to be an implication of his break with Plato moral ultra-realism. Aristotle’s agenda in his moral works is arguably dependent on a polemical intention, namely dismantling Socratic intellectualism. The semi-inductive or virtuously circular method of practical philosophy is illustrated, starting with the received opinions of the better and wiser individuals and trying dialectically to sift what is left of mistake and inconsistence in such opinions, finally trying to correct mistakes and make the overall practical science more consistent. The chapter illustrates then the relationship of individual ethics, or ‘monastics’, with the art and the science of the pater familias, or ‘economics’, and the science of the ruler and citizen, or politics. The nature of virtues, or better, excellences of character, is discussed, highlighting the basic role of hexis, or ‘disposition’. Prudence, or better practical wisdom, is the focus of the chapter. Its relationship with bouleusis or deliberation is examined, and its autonomous status vis-à-vis theoretical knowledge is stressed. -/- 3. Diogenes and philosophy as a way of life. The chapter provides an overview of Hellenistic ethics, which almost amounts to Hellenistic schools of philosophy, in so far as ‘philosophy’ became in these centuries primarily the name for a way of life. The typical character of the Cynical movement is highlighted, that of a school of life, not a school aimed at providing any kind of intellectual training was to be provided. -/- 4. Epicurus and ethics as self-care. The peculiar character of the Epicurean school is described, a combination of a science of well-being aiming, more than at pleasure as in the popular view, at reduction of useless suffering, of unnecessary needs, and at a balanced selection of pleasures of the best and most durable kind. -/- 5. Epictetus and ethics as therapy of the passions. The various phases of Stoicism are described, and the shifting place given to ethics in the Stoic system of idea, culminating in the paradoxical view of ethics, its impossibility in principle notwithstanding, as the only truly significant and necessary part of philosophy. Cicero is treated, showing how his own synthesis of various Hellenistic trends is as a truly philosophical enterprise, deserving serious consideration after one or two centuries when he was confined to the role of literate. Epictetus is chosen as the best example of what the Stoic tradition could yield, an art of living based on sophisticated introspection, in turn aimed at making a kind of cognitive therapy possible, dismantling obnoxious passions at their root by systematically correcting false representations. -/- 6. Philo and the reconciliation of Torah and Platonism. A reconstruction of basic ideas from a few books of the Hebrew Bible is provided, starting with the Prophetic tradition and the focus on God’s mercy as the source of motivation and standard for human behaviour. Then a comparative analysis is undertaken of a parallel tradition, namely the three codifications of the Torā (Law or, better, Instruction), highlighting how a core of moral ideas may be recognized as a basis and preamble of codification of civil law, cultural practice, and regulation of ritual purity. The importance of Leviticus is stressed as the turning point when emphasis mercy, typical of the Prophetic tradition, is combined with the legal tradition, yielding the change in sensibility of the Second Temple time. Philo of Alexandria is described as one of the three leading figures – together with Rabbi Hillel and Rabbi Yeshua – at the apex the emergence of the mentioned new sensibility, gradually including mercy as an essential part of justice and establishing the starting point of both the Rabbinic and the Christian tradition. This consists precisely in the precept of one’s neighbour’s love or of the golden rule, two eventually identical precepts whose meaning is arguably more sober and sensible than the long-lasting Christian tradition deriving from John and Augustine has made us believe and no novelty vis-à-vis so-called Ancient-Testament teachings. -/- 7. Augustine and Christianity as Neo-Platonism. The first section examines the moral doctrines of so-called 'ethical Treaties' from the Talmud, a group of treaties, among which the best known is Pirqé Avot, that were left out of the six "orders" of the canon as they did not fit in any of the six groups of issues ritual or legal on which the division was based. According to Maimonides, their peculiar theme is provided by the Deòt, 'opinions', i.e., mental dispositions, that is, the translation of the Greek term hexeis and Arab akhlak (in turn providing in this language the name for ethics as such). The three topics I reconstruct are: i) the notion of Torah: The Torah is understood as the world order itself, or as the ‘Wisdom’ that existed even before creation and was ‘the tool by which the world was built’; however, the Torah is an earthly and human entity, as it was "received" by humans, and from that very moment belongs to them; ii) the relationship between love of God and love of neighbour; the treaties require us to study and practice the Torah ‘for its own sake’, that is, require us to act out of love, not out of fear or hope of reward; iii) the idea of sanctification of daily life: having disappeared with the destruction of the Temple the possibility of any conflict between liturgical service and everyday life, the latter is assumed to be in itself divine service: to give food to the poor has the same value as sacrifices in the Temple, and as an implication, the insistence become recurrent on the goodness of created things in themselves along with a polemic against ascetic currents. The conclusions drawn are: i) the moral teachings of the Talmud and those of Yeshua are, rather than similar, virtually identical; one may safely say that the precept of love and the golden rule are central ones for all Talmudic rabbis, that mercy plays an indispensable role alongside with justice, and the latter is not a different thing from one’s neighbour’s love; ii) a peculiarity of Talmud rabbis facing Yeshua is the idea of study as worship, and knowledge as a source of justice; but this is an idea of Judaism after the Temple's destruction that cannot be attributed to the Pharisees of Yeshua’s time; iii) the relation of study and practice in the Talmud parallels that between faith and deeds in Paul's epistles, that is, respectively faith or learning are a necessary and sufficient condition to be recognized as righteous , but deeds are the inevitable effect of either faith or learning. The sayings ascribed to Yeshua are examined first, yielding the conclusion that a close equivalent may be found for every saying in Talmudic literature and yet the whole is ascribed to one rabbi, with rather consistent stress on God’s mercy and unconditional forgiving as the mark of true imitation of God. Thus, Yeshua’s teaching is pure Judaism. The third section describes briefly the galaxy of Gnostic currents and Manichaeism, trying to sketch the profile of moral teachings resulting from an encounter of Asian spiritual traditions, Hellenistic lore and sparks of teachings from apocalyptic Jewish currents. The last section summarizes the turbulent history of several encounters between Christian currents and Hellenistic philosophical schools. The first one was with late Cynicism. Recent, rather controversial, literature discovered the jargon and a few topics from the Stoic-Cynical popular philosophy in a few books from the New Testament itself. This, far from proving that Rabbi Yeshua had been influenced by cynic preachers, is a proof of the necessity felt by Christian preachers to translate the original ‘Christian’ teaching into a Greek lexicon deeply impregnated with cynic terminology. The second was with Platonism, yielding the mild and temperate moral teaching of Clemens of Alexandria, teaching the sanctity of nature and the human body, the joy of moderate fruition of ‘natural’ kinds of pleasures, and the beauty of the marital life – in short, the opposite of the standard picture of Medieval Christianity. Ambrose of Milan brings about a different kind of synthesis, namely with Middle Platonism, where Stoic themes prevail. The most shocking case is Augustine, where an early Manichean education is overcome in a former phase by a synthesis of Plotinian Neo-Platonism and Christian preaching, yielding a sustained polemic with the Manicheans and rather optimistic views on life and Creation, the body and sexuality, and Hebrew-Judaic tradition not far from Clemens of Alexandria. In a later phase, occasioned by controversy on the opposite front, with such Christian currents as the Pelagians and Donatists, Augustine comes back to heavily anti-Judaic and world-denying ascetic attitudes where is earlier Manichean upbringing seems to emerge again. The tragedy of medieval Christianity will be the later Augustine’s overwhelming influence. A final section is dedicated to the monastic tradition where a curious mixture of world-denying asceticism with an astonishingly penetrating ‘science’ of introspection emerges. -/- 8. Al Farabi and the reconciliation of Islam and Platonism. The Qurān and the qadit, that is, collections of sayings ascribed to the prophet Muhammad contain a wealth of precepts and catalogues of virtues mirroring moral contents from the Jewish and the Christian traditions, among them the basic notion of imitation of the Deity, where mercy is assumed to be its basic trait. An important tradition within Islam, namely Sufism, stressed to the utmost degree the role of mercy, turning Islam into a mystic doctrine centred on retreat from the world, abandonment to God’s will, and a peaceful and fraternal attitude to our fellow-beings. A secular literary tradition originating in the Sassanid Empire, the literature of advice to the Prince had for a time a widespread influence in the newly constituted Arabic-Islamic commonwealth. In a later phase, the legacy of Hellenic philosophy made its way into the intellectual elite of the Islamic world. The first important legacy was Platonic, and the Republic became the main text for Islamic Platonism. Al-Farabi wrote an enjoyable remake of the Platonic Republic, arguing that the citizen’s virtues were the basis on which the political building needs to rest. The secular lawgiver is enlightened by the light of reason in his everyday practice of governance, but room is made for the Prophet as the voice of revealed truth that confirms and completes the rational truth that philosophy may afford. In a second phase also Aristotelian and Stoic influence were assimilated. Miskawayh’s treatise was the masterwork at the time Arabic philosophy reached its Zenith. It is a treatment of the soul’s diseases and their remedies, combining the Aristotelian doctrine of the golden mean with the Stoic doctrine of the passions and elements of Galenic medicine. Towards the eleventh-twelve centuries a war raged among theologians and philosophers, finally won by the former with disappearance of philosophy as such. The newly established mainstream, yet, was no kind of intolerant fanaticism. It drew from the work of mystic theologian Al-Ghazālī, the best heir of Sufism, teaching a tolerant and peaceful attitude to our fellow-beings and a passive attitude to destiny as an expression of the Divine will. -/- 9. Moshe ben Maimon and the reconciliation of Torah and Aristotelian ethics. The encounter between the Talmudic tradition and Hellenic philosophy had taken place a first time in Alexandria at the time of Philo but the two traditions had parted way again. In fact, the kind of Platonic Judaism founded by Philo survived only within Christianity, in the fourth Gospel and then in writings by Clemens of Alexandria. The Talmudic literature had absorbed just less striking traits from the Hellenic Philosophy, namely an idea of ethics as care of the self and a role for the education of character as propaedeutic to theoretical knowledge. In the Arabic-Islamic world, a second round started when Jewish authors writing in Arabic undertook the task to prove the full compatibility of the tradition deriving from Torah and Platonic philosophy. The culmination of this attempt is provided by Moshe ben Maimon who tried to use Aristotelian ethics as a language into which the teaching from the Pirké Avot could be translated. -/- 10. Thomas Aquinas and the reconciliation of Christian morality and Aristotelian ethics. In the first section the fresh start is described of philosophical ethics in Latin Europe at the turn of the Millennium. In a first phase, Anselm and Peter Abelard built on a Platonic and Augustinian legacy. In this phase. remarkable innovations are introduced, including Abelard’s claim of the obliging character of erring consciousness. In a second phase, the rediscovery of Nicomachean Ethics thanks to Latin translation of Arabic versions gives birth to a new wave of ethical studies, recovering the very idea of Aristotelian practical philosophy, with the potential implication of full legitimization of natural morality, i.e. ethics without Revelation. Aquinas’s ethics is a theological ethics out of which it would be vain to try to extract a self-contained philosophical ethics. His treatment of topics in philosophical ethics, yet, does not boil down to repetition of Aristotelian arguments but is rather a creative reshaping of such arguments. For example, he introduces also in the practical philosophy a first principle parallel to the principle of non-contradiction; and he also carries out a synthesis of Aristotelianism, Stoicism, and neo-Platonism. Even though it is essentially moral theology, Aquinas’s doctrine - unlike Augustine – grants full citizenship to "natural" morality, firstly by rejecting the claim that the corruption of human nature due to the original sin is so radical as to leave "pure nature" incapable of moral goodness. The doctrine is presented in a more sophisticated formulation in a few of the Quaestiones, such as De Malo and De Veritate, in the Summa contra Gentes and in the commentaries to Aristotle than in the famous Summa Theologica, but the latter work includes the only or the largest exposition of some decisive part of the theory. Thus, the Summa Theologica should be read for what it is more than criticized for not being what it was not meant to be. It was not meant to be the brilliant synthesis of all that Reason he had been able to produce with what Revelation had added about which the Neo-Thomists used to dream, but rather a manual for the training of preachers and confessors, where theoretical claims are not too demanding and a few serious tensions are left. Besides a jump between the Prima Secundae and the Secunda Secundae, being the former an essay in virtue theory and the latter a handbook for confessors, the most serious tension is perhaps the one between the ethics of right reason presented in most of Ia-IIae and the eudemonistic ethics developed in quaestiones 1-5 of the same part; the alternative ethical theory which also may be found in Augustine, the Stoic view of a cosmic reason eventually coincident with the moral law, was believed by Anselm (followed by John Duns Scotus and William of Ockham) to be incompatible with eudemonism. It is questionable whether Thomas could reduce the tension proving that it is just an apparent tension, in so far as the right reason and bliss derived from knowledge of God tend to coincide, but this is just a conjecture. Thomas’s ethics is a virtue ethic, not a law-based one, and moral judgment focuses on the virtues, particularly charity, not on the commandments and even less on absolute prohibitions; Thomas, however, would not have considered a drastic alternative between law and the virtues such as the one which has been advanced in late twentieth-century philosophy to be justified. Nonetheless, when it discusses ‘special’ virtues, it ceases to be an ethics of virtue and becomes a disappointing and often contradictory discussion of legal and illegal acts. Such a discussion takes most of the time ‘reasonable’ middle positions on controversial issues but not the alternative approach that Aristotelianism would have made possible; even when some occasional Aristotelian claim shows up, such as money’s barrenness as a reason against usury, this seems to be made by an author who apparently ignores the Aristotelian Thomas of the Prima Secundae. It is an ethic of human autonomy which recognizes the binding character of the individual conscience and, potentially, even a duty to disobey unjust laws. It is true that what Thomas writes in his discussion of death penalty and persecution of heretics is simply disgusting, and yet we should blame Thomas the man, not Thomist ethical theory. Finally, Thomas’s ethics is not ethical ‘absolutism’, as both traditionalist Catholic theologians and secularist enemies of dogmatic morality tend to believe. It is instead an ethic of prudential judgment. Exceptions to this approach – or better results of logical fallacies – are such claims as the absolute character of negative precepts and of the existence of "intrinsically evil acts", claims that simply cease to make sense in the light of Thomas’s own distinction between human act and natural act, carrying consequences Thomas did not live long enough to draw or was not consistent enough to dare to draw. -/- 11. Francisco de Vitoria and casuistry. The first topic illustrated is the discussion about the notion of pura natura, namely the human condition after the original sin and before divine revelation. The core notion was already there in Aquinas but was later developed by Caietanus (Thomas de Vio) with a number of interesting implications: firstly a view of human nature as such much more positive than Augustine’s and his followers’ view, including both Calvinists and Jansenists; secondly, the implication that the philosopher’s morality, as opposed to the Christian’s morality, is a quite respectable kind of morality; thirdly, that theocracy and prosecution of the unfaithful lacked justification, with more far-fetching consequences than Aquinas himself had dared to draw. The second topic is casuistry, a new name for a comparatively older way of thinking, which was already there in late Stoicism and Cicero as well as in the Talmudic literature. This is an approach aimed at yielding probable enough solutions for doubtful cases even in absence of completely safe staring points. The genre developed from medieval reference books for confessors and became by the sixteenth-century a sophisticated tool-box for dealing with various fields of applied ethics. Francisco Vitoria, the main authority of Baroque Scholasticism, was a controversial figure, among the proponents of the new discipline of casuistry and a consistent proponent of more separation between the religious and the civil authority, stricter limits to the legitimacy of war, innate rights of non-Christian nations in the New World based on the notion of pura natura, providing a standard of ‘natural’ goodness, previous to revelation, on which the Indian nations were judged to live in a naturally good condition, provided with the institutions of family, justice, and religion had accordingly a right to full respect by Christian sovereigns. Bartolomé de Las Casas, arguing along similar lines, was the leader of a historical battle in defence of the rights of the Indios. -/- 12. Michel de Montaigne and the art of living. A short description of one of the Renaissance Phyrronism, one of the classical philosophies revived in the fifteenth and sixteenth century. Montaigne combines the sceptical epistemology with an understanding of ethics – indeed of philosophy as a whole – in terms of an art of living inspired by two basic ideas, sagesse, that is, practical wisdom as the only kind of rationality available after theology, science, and tradition have declared bankrupt, and an aesthetic ideal of self-transformation through the art of writing and introspection. -/- 13. Pierre Nicole and neo-Augustinianism. The chapter illustrates first the vicissitudes of Augustinianism newly born after the late medieval triumph of Thomist Aristotelianism in the alternative Protestant and Catholic versions processed by the Calvinists and the Jansenists. Then it illustrates the doctrines of Pierre Nicole, Blaise Pascal’s best disciple, on the impossibility of introspection, on the ubiquity of self-deception, on the incurably evil character of the passion of love, and on the two moralities, the one of the man of the world, the morality of honesty which is indispensable for granting peace and order but devoid of any true value for eternal salvation, since the same external behaviour may be prompted by opposite motivations, and the morality of charity, the only true one but also useless to society. The third topic is Pierre Malebranche’s view of self-deception as a ubiquitous phenomenon accounting for human action and responsibility and his reformulated theory of self-love, no less ubiquitous than for Nicole and yet not as incurably evil, given the distinction between morally indifferent and even necessary amour de soi and vicious amour proper, a distinction on which the whole of Rousseau’s moral and political theory rests. -/- 14. Samuel von Pufendorf and the new science of morality. The chapter is dedicated to the discovery of the idea of a ‘new moral science’. Hugo Grotius is discussed first highlighting the real character of his project, much closer to the Thomist idea of a natural law accessible to non-corrupted human reason when human beings are living in a state of pura natura. Then Hobbes’s combination of scepticism, voluntarism, and conventionalism is described and both the continuity with Grotius’s new science, in the search for non-revealed rational morality, and the break with him, in the adoption of voluntarism and refusal of an intellectualist view of natural law, are illustrated. Pufendorf’s work is illustrated as a synthesis of the two previous attempts and the – up to Schneewind underestimated – paradigmatic example of the new science of morality. -/- 15. Richard Cumberland and consequentialist voluntarism. The chapter gives an overview of eighteenth-century Anglican ethics, noticing how the Cambridge tradition gave special weight to natural theology as opposed to positive or revealed theology – and how two Cambridge fellows, John Gay and Thomas Brown, elaborated on Cumberland’s (and Malebranche’s, as well as Leibniz’s) strategy for finding a third way between intellectualist view and voluntarist view of the laws of nature. The result of their elaboration was a kind of a rational-choice account of the origins of natural laws, where a law-giver God chooses among a number possible sets of laws on a maximizing criterion, and God’s maximandum is happiness for his creatures. The chapter notices also how such a solution aimed at solving at once the problem of evil and that of the foundation of moral obligation by proving how God’s choice was justified as far as it was the one minimising the amount of suffering in the world. 16. Richard Price and intuitionism. The chapter describes first the doctrines of the Cambridge Platonists, an example of hyper-rationalist reaction to Calvinism. Secondly it describes the sophisticated and universally ignored – from Sidgwick to Anscombe –version of what was later labelled ‘ethical intuitionism’ – showing how it escapes familiar objections and misrepresentations of intuitionism – from Mill to Rawls – in grounding its argument on transcendental arguments while carefully avoiding recourse to introspection and psychological evidence, which has been taken as a too easy target by critics of intuitionism. Thirdly, the chapter discusses Whewell’s ‘philosophy of morality’, as opposed to ‘systematic morality’, not unlike Kant’s distinction between a pure and an empirical moral philosophy. Whewell worked out a systematization of traditional normative ethics as a first step before its rational justification; he believed that the point in the philosophy of morality is justifying a few rational truths about the structure of morality such as to rule hedonism, eudemonism, and consequentialism; yet a system of positive morality cannot be derived solely from such rational truths but requires consideration of the ongoing dialectics between idea and fact in historically given moralities. Whewell’s intuitionism turns out closer to Kantian ethics than commentators have made us believe until now, and quite different from what Sidgwick meant by intuitionism. -/- 17. Adam Smith and the morality of role-switch. The chapter describes first, Hutcheson’s attempt at basing the ‘new science’ of natural law on different bedrock than Pufendorf and the English Platonists, namely a moral sense, a faculty whose existence is assumed to be proved on an empirical basis. The second step is a reconstruction of Hume’s rejoinder in terms of a new science of man including morality on an ‘experimental’ basis, that is, a ‘Newtonian’ hypothetical-deductive approach, distinguished from Hutcheson’s allegedly uncritical descriptive approach to human nature. The third step is a reconstruction of Adam Smith theory of morality understood as emerging from a spontaneous interplay of exchanges of situations (the most basic meaning of the word ‘sympathy’ as construed by this author). -/- 18. Jeremy Bentham and utilitarianism. The bulk of the chapter is dedicated to the doctrines of Jeremy Bentham. The Enlightenment spirit that suggested the idea of a new morality, free from religion, is illustrated. The notion of utility is illustrated as well as the subsequent formulations of the principle of utility. The idea of felicific calculus is discussed, showing how its inner difficulties prompted several reformulations of the principle of utility in order to avoid undesirable implications of proposed formulations. The role of the thesis of spontaneous convergence between interest and duty is discussed, showing how it left numberless questions open, and the distinction between the virtues of prudence, justice and beneficence is described as a way out of the deadlocks of classical utilitarianism. After Bentham, Mill’s reformed utilitarianism is reconstructed, showing how it is a kind of mixed system – as closer to common sense as it gives up Bentham’s claims to consistency and simplicity – resulting as unintended consequences from the controversies in which Mill was too keen to engage. The hidden agenda of the controversy between Utilitarianism and Intuitionism, going beyond the image of the battle between Prejudice and Reason, is described, showing how both competing philosophical outlooks turn out to be more research programs than self-contained doctrinal bodies, and such programs appear to be implemented, and indeed radically transformed while in progress thanks to their enemies no less than to their supporters. -/- 19. Immanuel Kant, practical reason and judgment. The chapter argues that Kant took from Moses Mendelssohn the idea of a distinction between geometry of morals and a practical ethic. He was drastically misunderstood by his followers precisely on this point. He had learned from the sceptics and the Jansenists the lesson that men are prompted to act by deceptive ends, and he was aware that human actions are also empirical phenomena, where laws like the laws of Nature may be detected. His practical ethics made room for judgment as a holistic procedure for assessing the saliency of relevant moral qualities in one given situation; this procedure yields the same results as the geometry of morals does for abstract cases but does so immediately and without balancing conflicting duties with each other, since what makes for the salient quality of a situation is perceived from the very beginning. Kant's practical philosophy is richer than the received image, making room for an ‘empirical moral philosophy’ or moral anthropology including treatment of commerce, needs, value and price, happiness and well-being; the overall social theory and philosophy of history is less different from Adam Smith's than the received image makes believe; the paradox of happiness is central to Kant's philosophy; a distinction between happiness and well-being is clearly drawn; the distinction plays a basic role in establishing a link between the economic and the moral life. -/- 20. Georg Wilhelm Friedrich Hegel and the critique of abstract universalism. The chapter describes first the Romantic movement and the implications some of its concerns, rescuing passions, community, tradition, the individual as the bearer of a unique destiny, and the longing for organic unity between the individual, mankind and nature. Hegel’s contribution is discussed then, highlighting how, on the one hand, he shared a number of these concerns and on the other he had more rationalist leanings. The notion of morality is the pivotal point of the reconstruction, highlighting how Hegel construes this notion as a key to his own diagnosis of the malaise of modernity – the separation of individual and Gemeinschaft – and how his attack on Kant turns around this very idea. -/- 21. Friedrich Nietzsche against Christianity and the Enlightenment. The chapter illustrates first the idea of deconstruction of the back-world of values. Nietzsche claims to be legitimate heir of the French moralists, Leibniz, Kant and Hegel, in so far as he allegedly carries out to its deepest implications their discovery of what lies behind traditional naïve belief in the existence of an objective realm of values just waiting for description by the philosopher. The two exemplars form which genealogy draws inspiration are the classical philologist’s historical reconstructions of lost meanings and the chemist’s decomposition of elements. Then the genealogy of the notions of good and evil carried out in the first dissertation of Genealogy of morals is illustrated with its paradoxical conclusion that will to power is in fact the only ‘genuine’ kind of goodness. The third point illustrated is the dialectic of ascetism and self-realization with its ambiguous outcome. The suggested interpretation of such outcome is that Nietzsche’s normative ethics is a kind of virtue ethics taking an aesthetic ideal as a normative standard -/- 22. George Edward Moore and ideal utilitarianism. The chapter discusses the ideas of common sense and common-sense morality in Sidgwick. I argue that, far from aiming at overcoming common-sense morality, Sidgwick aimed purposely at grounding a consist code of morality by methods allegedly taken from the natural sciences to reach, also in ethics, the same kind of “mature” knowledge as in the natural sciences. His whole polemics with intuitionism was vitiated by the a priori assumption that the widespread ethos of the educated part of humankind, not the theories of the intuitionist philosophers, was what was worth considering as the expression of intuitionist ethics. In spite of a naïve positivist starting-point, Sidgwick was encouraged by his own approach in exploring the fruitfulness of coherentist methods for normative ethics. Thus, Sidgwick left an ambivalent legacy to twentieth-century ethics: the dogmatic idea of a “new” morality of a consequentialist kind, and the fruitful idea that we can argue rationally in normative ethics albeit without shared foundations. Then it reconstructs the background of ideas, concerns and intentions out of which Moore’s early essays, the preliminary version, and then the final version of Principia Ethica originated. I stress the role of religious concerns, as well as that of the Idealist legacy. I argue that PE is more a patchwork of rather diverging contributions than a unitary work, not to say the paradigm of a new school in Ethics. I add a comparison with Rashdall’s almost contemporary ethical work, suggesting that the latter defends the same general claims in a different way, one that manages to pave decisive objections in a more plausible way. I end by suggesting that the emergence of Analytic Ethics was a more ambiguous phenomenon than the received view would make us believe, and that the wheat (or some other gluten-free grain) of this tradition, that is, what logic can do for philosophy, should be separated from the chaff, that is, the confused and mutually incompatible legacies of Utilitarianism and Idealism. -/- 23. Edmund Husserl and the a-priori of action. The chapter illustrates first the idea of phenomenology and the Husserl’s project of a phenomenological ethic as illustrated in his 1908-1914 lectures. The key idea is dismissing psychology and trying to ground a new science of the a priori of action, within which a more restricted field of inquiry will be the science of right actions. Then the chapter illustrates the criticism of modern moral philosophy carried out in the 1920 lectures, where the main target is naturalism, understood in the Kantian meaning of primacy of common sense. The third point illustrate is Adolph Reinach’s theory of social acts as a key the grounding of norms, a view that basically sketches the very ideas ‘discovered’ later by Clarence I. Lewis, John Searle, Karl-Otto Apel and Jürgen Habermas. A final section is dedicated to Nicolai Hartman, who always refused to define himself a phenomenologist and yet developed a more articulated and detailed theory of ‘values’ – with surprising affinities with George E. Moore - than philosophers such as Max Scheler who claimed to Husserl’s legitimate heirs. -/- 24. Bertrand Russell and non-cognitivism. The chapter reconstructs first the discussion after Moore. The naturalistic-fallacy argument was widely accepted but twisted to prove instead that the intuitive character of the definition of ‘good’, its non-cognitive meaning, in a first phase identified with ‘emotive’ meaning. Alfred Julius Ayer is indicated as a typical proponent of such non-cognitivist meta-ethics. More detailed discussion is dedicated to Bertrand Russell’s ethics, a more nuanced and sophisticated doctrine, arguing that non-cognitivism does not condemn morality to arbitrariness and that the project of a rational normative ethics is still possible, heading finally to the justification of a kind of non-hedonist utilitarianism. Stevenson’s theory, another moderate version of emotivism is discussed at some length, showing how the author comes close to the discovery of the role of a pragmatic dimension of language as a basis for ethical argument. Last of all, the discussion from the Forties about Hume’s law is described, mentioning Karl Popper’s argument and Richard Hare’s early non-cognitivist but non-emotivist doctrine named prescriptivism. -/- 25. Elizabeth Anscombe and the revival of virtue ethics. The chapter discusses the three theses defended by Anscombe in 'Modern Moral Philosophy'. I argue that: a) her answer to the question "why should I be moral?" requires a solution of the problem of theodicy, and ignores any attempts to save the moral point of view without recourse to divine retribution; b) her notion of divine law is an odd one more neo-Augustinian than Biblical or Scholastic; c) her image of Kantian ethics and intuitionism is the impoverished image manufactured by consequentialist opponents for polemical purposes and that she seems strangely accept it; d) the difficulty of identifying the "relevant descriptions" of acts is not an argument in favour of an ethics of virtue and against consequentialism or Kantian ethics, and indeed the role of judgment in the latter is a response to the difficulties raised by the case of judgment concerning future action. A short look at further developments in the neo-naturalist current is given trough a reconstruction of Philippa Foot’s and Peter Geach’s critiques to the naturalist-fallacy argument and Alasdair MacIntyre’s grand reconstruction of the origins and allegedly unavoidable failure of the Enlightenment project of an autonomous ethic. -/- 26. Richard Hare and neo-Utilitarianism. The chapter addresses the issue of the complex process of self-transformation Utilitarianism underwent after Sidgwick’s and Moore’s fatal criticism and the unexpected Phoenix-like process of rebirth of a doctrine definitely refuted. A glimpse at this uproarious process is given through two examples. The first is Roy Harrod Wittgensteinian transformation of Utilitarianism in pure normative ethics depurated from hedonism as well as from whatsoever theory of the good. This results in preference utilitarianism combined with a ‘Kantian’ version of rule utilitarianism. The second is Richard Hare’s two-level preference utilitarianism where act utilitarianism plays the function of eventual rational justification of moral judgments and rule-utilitarianism that of action-guiding practical device. -/- 27. Hans-Georg Gadamer and rehabilitation of practical philosophy. The post-war rediscovery of ethics by many German thinkers and its culmination in the Sixties in the movement named ‘Rehabilitation of practical philosophy’ is described. Among the actors of such rehabilitation there were a few of Heidegger’s most brilliant disciples, and Hans-Georg Gadamer is chosen as a paradigmatic example. His reading of Aristotle’s lesson I reconstructed, starting with Heidegger’s lesson but then subtly subverting its outcome thanks to the recovery of the central role of the notion of phronesis. -/- 28. Karl-Otto Apel and the revival of Kantian ethics. Parallel to the neo-Aristotelian trend, there was in the Rehabilitation of practical philosophy a Kantian current. This started, instead than the rehabilitation of prudence, with the discovery of the pragmatic dimension of language carried out by Charles Peirce and the Oxford linguistic philosophy. On this basis, Karl-Otto Apel singled out as the decisive proponent of the linguistic and Kantian turn in German-speaking ethics, worked out the performative-contradiction argument while claiming that this was able to provide a new rational and universal basis for normative ethics. An examination of his argument is some detail is offered, followed by a more cursory reconstruction of Jürgen Habermas’s elaboration on Apel’s theory. -/- 29. John Rawls and public ethics as applied ethics. Rawls’s distinction between a “political” and a “metaphysical” approach to one central part of ethical theory, namely the theory of justice, is interpreted as a formulation of the same basic idea at the root of both the principles approach and neo-casuistry, both discussed in the following chapter, namely that it is possible to reach an agreement concerning positive moral judgments even though the discussion is still open – and in Rawls’ view never will be close – on the basic criteria for judgment. -/- 30. Beauchamp and Childress and bioethics as applied ethics. The chapter presents the revolution of applied ethics while stressing its methodological novelty, exemplified primarily by Beauchamp and Childress principles approach and then by Jonsen and Toulmin’s new casuistry. -/- . (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Philosophy and Literature 25.1 (2001) 1-13 [Access article in PDF] From Postmodernism to Postmodernity: The Local/Global Context Ihab Hassan I What Was Postmodernism? What was postmodernism, and what is it still? I believe it is a revenant, the return of the irrepressible; every time we are rid of it, its ghost rises back. Like a ghost, it eludes definition. Certainly, I know less about postmodernism today than I did (...) thirty years ago, when I began to write about it. This may be because postmodernism has changed, I have changed, the world has changed.But this is only to confirm Nietzsche's insight, that if an idea has a history, it is already an interpretation, subject to future revision. What escapes interpretation and reinterpretation is a Platonic Idea or an abstract analytical concept, like a circle or a triangle. Romanticism, modernism, postmodernism, however, like humanism or realism, will shift and slide continually with time, particularly in an age of ideological conflict and media hype.All this has not prevented postmodernism from haunting the discourse of architecture, the arts, the humanities, the social and sometimes even the physical sciences; haunting not only academic but also public speech in business, politics, the media, and entertainment industries; haunting the language of private life styles like postmodern cuisine--just add a dash of raspberry vinegar. Yet no consensus obtains on what postmodernism really means.The term, let alone the concept, may thus belong to what philosophers call an essentially contested category. That is, in plainer language, if you put in a room the main discussants of the concept--say Leslie Fiedler, Charles Jencks, Jean-François Lyotard, Bernard Smith, Rosalind Krauss, Fredric Jameson, Marjorie Perloff, Linda Hutcheon and, just to [End Page 1] add to the confusion, myself--locked the room and threw away the key, no consensus would emerge between the discussants after a week. But a thin trickle of blood might appear beneath the sill.Let us not despair: though we may be unable to define or exorcise the ghost of postmodernism, we can approach it, surprising it from various angles, perhaps teasing it into a partial light. In the process, we may discover a family of words congenial to postmodernism. Here are some current uses of the term: 1. Frank Gehry's Guggenheim Museum in Bilbao (Spain), Ashton Raggatt McDougall's Storey Hall in Melbourne (Australia), and Arata Isozaki's Tsukuba Center (Japan) are considered examples of postmodern architecture. They depart from the pure angular geometries of the Bauhaus, the minimal steel and glass boxes of Mies van der Rohe, mixing aesthetic and historical elements, flirting with fragments, fantasy, and even kitsch. 2. In a recent encyclical, titled "Fideset Ratio," Pope John Paul II actually used the word postmodernism to condemn extreme relativism in values and beliefs, acute irony and skepticism toward reason, and the denial of any possibility of truth, human or divine. 3. In cultural studies, a highly politicized field, the term postmodernism is often used in opposition to postcolonialism, the former deemed historically feckless, being unpolitical or, worse, not politically correct. 4. In Pop culture, postmodernism--or PoMo as Yuppies call it insouciantly--refers to a wide range of phenomena, from Andy Warhol to Madonna, from the colossal plaster Mona Lisa I saw advertising a pachinko parlor in Tokyo to the giant, cardboard figure of Michelangelo's David--pink dayglo glasses, canary shorts, a camera slung across bare, brawny shoulders--advertising KonTiki Travel in New Zealand. What do all these have in common? Well, fragments, hybridity, relativism, play, parody, pastiche, an ironic, anti-ideological stance, an ethos bordering on kitsch and camp. So, we have begun to build a family of words applying to postmodernism; we have begun to create a context, if not a definition, for it. More impatient or ambitious readers can consult Hans Bertens' The Idea of the Postmodern, the best and fairest [End Page 2] introduction I know to the topic. But now I must make my second move or feint to approach postmodernism from a different perspective. II Postmodernism/Posmodernity. I make this move by... (shrink)

The essay consists in the analysis of the problem of the evil in the man and in the analysis of the remedies which the man can find against the evil. Plato affirms the presence of an active principle of evil in the soul of every man, which coincides with some instincts of the appetitive soul; the opposite principle to the evil is the reason, which needs, though, a correct education in order to be able to fight efficiently against the evil (...) in us. The man can be seen as a battle field of these opposite forces. Plato describes the presence of the evil in us in some passages of Republic Book 9, where he compares the appetitive part of the soul with a monster. The destiny of every person in her earthly existence consists in the continuing control of the appetitive part of the soul, if the status of ethical education is to be reached and maintained. The man who remains in the realm of the opinion, that is, in the realm of the doxa is an individual who only disposes of unstable opinions and who as a consequence do not have authentic remedies against the appetitive part. On the contrary, the individual who can ascend to the realm of being through the hard education represented by arithmetic, geometry, stereometry, astronomy, harmony and, finally, dialectic is really able to contrast the force of the evil within the individual. Ethics is really possible only through the complete education which passes through these disciplines: the more the individuals is theoretically educated, the more the individual is ethically educated. The knowledge of ideas is the only authentic therapy against the evil in us. (shrink)

The volume starts with the paper of Lynn Maurice Ferguson Arnold, former Premier of South Australia and former Minister of Education of Australia, concerning the Exposition Internationale des Arts et Techniques dans la Vie Moderne (International Exposition of Art and Technology in Modern Life) that was held from 25 May to 25 November 1937 in Paris, France. The organization of the world exhibition had placed the Nazi German and the Soviet pavilions directly across from each other. Many papers are devoted (...) to the interpretation of this opposition. Arnold’s paper considers the differences in the two totalitarian states’ architectural and design discourse styles. Although each of them communicated a totalitarian language of purposes, permissions, and boundaries, they essentially differed in the styles of discourse represented by the architecture and design of their respective pavilions. They were opposites of each other and the liberal ideals they contested. The internationalist viewpoint reflecting the multi-ethnic mix of the USSR is contrasted to the ein Volk homogeneity represented in the German pavilion. The Soviet pavilion opted for a utopian future to be arrived at by “benign” leadership, whereas the German pavilion anchored itself in the myth of Teutonic history with the nostalgic pride protected by the swastika-bearing eagle. Jean-Yves Béziau is best known as a logician and founder of Universal Logic. However, in this paper, he poses a new philosophical question: why philosophical discourse does not use images? The paper begins with a general analysis of different types of philosophical discourses. Then, he focuses on why images have been and are still rejected in philosophical discourse. He explains various ways to use images in a fruitful way to develop philosophical thinking and discourse and illustrates his view by providing several examples. He concludes with a promising programmatic declaration about founding a new journal entitled World Journal of Pictorial Philosophy to stimulate the usage of images for developing philosophical thinking. Although complaints about the obscurity of many philosophers’ discourse are widespread, Tatiana Denisova, ex-professor of the Surgut University and a research associate of the University of the Aegean undertakes a positive attitude to this problem. She explains that the reasons for the obscurity of philosophical texts, the subsequent complexity in communicating philosophers’ meanings and their eventual incomplete understanding is not a sign of their inferiority. On the contrary, it is a sign of the fruitfulness of philosophical discourse, which can generate new meanings. Thus, the darkness of philosophical discourse is like the life-giving chaos, and the obscurity that it inevitably contains can be the keeper of implicit meanings and even their generator. Katarzyna Gan-Krzywoszyńska and Piotr Leśniewski, authors of a monograph on Polish philosopher Kazimierz Ajdukiewicz (1890 – 1963), make a comparative analysis of his educational style with that of Paulo Reglus Neves Freire (1921 – 1997). The authors highlight that these scholars have distinctly different backgrounds. The former was an educator, philosopher and a leading advocate of critical pedagogy connected with the dialogical Latin-American tradition. The latter was a philosopher and logician, a notable representative of the Lwów–Warsaw school of logic with significant contributions to semantics, model theory and the philosophy of science. Despite their apparent difference, the authors identify some striking similarities regarding their attitudes towards education; notably, their approaches are essentially dialogical. Gilah Yelin Hirsch is a multidisciplinary artist who works as a painter, writer, curator, educator, and filmmaker. In her experiential paper, he explains how and why she uses four channels of communication in her creative expression: writing, painting, filmmaking, and teaching, as dialogic inquiry. She considers that these channels compose a continuum of discourse styles, each reaching a different facet of kaleidoscopic consciousness. She claims that her choice of medium is prompted by a need to communicate her insights profoundly. Thus, for instance, she created painted allegories to express narratives based on dreams and visions emerging from her subconscious. Art and healing is another Tibetan-rooted type of discourse practised by the author. Creating and observing a healing image produces a positive psychophysiological change in both the artist and the viewer. The discourse here is tripartite between creator, image, and viewer. Filmmaking is another type of discourse she practices between the filmmaker/artist as shaman or healer and the viewer as respondent or participant. These films are meant to be experienced frame by frame, physically and emotionally in both the body and mind. Jocelyn Ireson-Paine is a cartoonist and programmer. His paper offers an original attempt to use category theory in cartooning. Category theory is a branch of mathematics that provides concepts and methods to describe general abstract structures in terms of a labelled directed graph called category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms or transformations or mappings). The author explores different ways to define “style” in art using category theory and a relational view of art. Moreover, he examines how “translation” (transformation) between styles could be defined and reveals the difficulties of how it could be implemented in a computer using special software. Jens Lemanski is a philosopher who has revived research in Arthur Schopenhauer’s legacy by exploring his work on mathematical evidence, logic diagrams, and problems of semantics. In this paper, he advances a new approach to eristic as an art of protecting oneself from the one who deliberately violates norms of discourse ethics to win an argument. The author attributes the origins of this view to Schopenhauer, suggesting a new reading of his work. According to the author, eristic is a prohibitive technique that takes effect when the norms of discourse ethics are transgressed and violated. Thus, eristic is viewed as a discipline of Enlightenment philosophy and a correlate of discourse ethics. Roshdi Rashed is an authority on the history of Arabic mathematical sciences. Proceeding from Gilles-Gaston Granger’s definition of mathematical style, he studies the question of whether a mathematical work can be characterized by a single style or by a multitude of styles. This question is explored within the style of a single work, namely Menelaus’s Sphaerica, and through the study of the development of a single problem over time, namely the isoperimetric problem. He indicates Menelaus’s divergence from the Euclidean style of (plane and stereometric) geometry caused by the drop of the Parallel Postulate, which associated it with hyperbolic geometry. Hence, the author concludes that Menelaus’s Sphaerica incorporates the well-known Euclidean style with a variation of the non-Euclidean one. On the other hand, examining the isoperimetric problem reveals another interesting historical picture. A succession of different styles (cosmological, geometric, infinitesimalistic, style of the calculus of variations, of synthetic geometry) can be observed due to the transformation of the research object over time. Boris Shalyutin, a Russian social philosopher, and Ombudsman for Human Rights in the Kurgan Region, explores the origins of discourse in combination with the birth of society and law. He claims that legal discourse marks the historical emergence of discourse in general. Homo Juridicus generated Homo Sapiens, which have created new spheres of discourse, notably moral discourse and further philosophical, political, and scientific discourse. Petros Stefaneas, a logician, computer scientist and novelist, suggests an alternative to the traditional narratology approach for studying (interactive) social media discourse. He claims that style describes how the parts of a narrative are blended into a whole. The author relies upon Goguen’s and Harrel’s concept of style as a choice of blending principles and transfers to the study of the social media narratives elements from the methodology of studying Web-based collaborative search for mathematical proofs like those implemented within the Polymath project. The last paper belongs to Ghil‘ad Zuckermann, a linguist, language revivalist, and proponent of a model of the emergence of Israeli Hebrew, according to which Hebrew and Yiddish were the primary sources of Modern Hebrew. This paper explores the fascinating and multifaceted Yiddish language and its survival in Israeli. Yiddish is characterized by a unique style that embeds psycho-ostensive expressions throughout its discourse. The author highlights the cross-fertilization between Hebrew and Yiddish as it manifests itself in any aspect of the Israeli language. He claims that Yiddish survives beneath Israeli phonetics, phonology, discourse, syntax, semantics, lexis, and even morphology, although traditional and institutional linguists have been most reluctant to admit it. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 62.2 (2001) 193-210 [Access article in PDF] Zeno Against Mathematical Physics Trish Glazebrook Galileo wrote in The Assayer that the universe "is written in the language of mathematics," and therein both established and articulated a foundational belief for the modern physicist. 1 That physical reality can be interpreted mathematically is an assumption so fundamental to modern physics that chaos and super-strings are examples (...) of physical theories developed on the basis of success in mathematics. The mathematics came first, then the physical theory. Quantum theory is an awkward and similar case in point. Despite the fact that there is much agreement among physicists about the mathematical theory, its physical interpretation remains a matter of controversy. There are several interpretations, all of which challenge our everyday assumptions about reality. This led Bohr in 1935 to call for "a radical revision of our attitude towards the problem of physical reality." 2 Arthur Fine, originally a staunch defender of realist interpretations of quantum theory, saw the success of Bohr's Copenhagen interpretation as the end of Einsteinian realism, and subsequently Fine declared that "realism is well and truly dead." 3 Mathematical operators simply do not correspond with real entities in any conceivable way.Ilkka Niiniluoto has argued convincingly, however, that realism is alive and well in quantum theory. 4 The problem is that Bohr is right: quantum theory can be considered to concern the real only upon radical revision of ordinary conceptions of reality. Quantum theory does not describe the real in any meaningful sense of "the real." Rather, it is mathematical projection: ideal rather than real. [End Page 193]Aristotle did not hold that the physicist deciphers nature mathematically. He distinguished the physicist from the mathematician at Physics 2.2 by analogy to the definitions of "snub nose" and "curved." 5 The definition of "snub nose" requires the mention of a nose, that is, of matter which is inseparable from motion, whereas "curved" and other mathematical concepts like " 'odd' and 'even', 'straight'... 'number,' 'line,' and 'figure,' " which are separable from matter and from motion, are investigated by the mathematician. 6 Aristotle claims that physics is different from mathematics because the mathematician investigates physical lines, but not qua physical, and the physicist investigates mathematical lines "but qua physical, not qua mathematical." 7A peculiar consequence of my argument is that it establishes a common principle between two thinkers whose beliefs are otherwise so fundamentally in opposition. Zeno's paradoxes are commonly read as a denial of motion. For Aristotle, that move is axiomatic to the physicist. At 185a15 he considers arguments against this assumption outside the realm of objections to which the physicist must reply. If I am right that Zeno's paradoxes of motion challenge the very notion of mathematical physics, then he and Aristotle agree, contra Galileo, that the universe is not written in the language of mathematics. They simply come to this point from opposite directions, Zeno from an Eleatic thesis that physical reality is illusion, Aristotle from the natural scientist's pragmatic commitment to a truth about nature.I intend to use Zeno's paradoxes of motion to argue that the application of mathematical concepts to the physical world results in paradox. Zeno's arguments are limited within mathematics to geometry. This paper is accordingly a beginning from which I hope more work can be done later. Furthermore, Zeno's paradoxes are standardly read as directed against anyone who disagrees with Parmenides' thesis that there is but one thing which is motionless and indivisible. The paradoxes achieve this end by means of a reductio against the possibility of dividing space and time. That there are four paradoxes is accordingly explicable by the fact that there are four possibilities for the divisibility of space and time. The Achilles argues against the possibility that space and time are both infinitely divisible; the Dichotomy, against the infinite divisibility of space and finite divisibility of time; the Arrow, against the infinite divisibility of time and the finite divisibility of space; and finally, the Stadium shows that time and space cannot both... (shrink)

This paper begins by pointing to an obvious difficulty in Merleau-Ponty’s late philosophy: undoing the decisive separation between linguistic connotation and the denotated, undoing the decisive separation between linguistic meaning and the sensible world. This difficulty demands that we understand how the sensible and the symbolic have a sort of spontaneous relation. How can this be? The history of this problem is then traced back to Husserl, and in particular to his The Origin of Geometry. For Husserl, ‘abstract (...) class='Hi'>geometry’ is understood to be a self-constituting field that cuts itself off from a ‘pre-geometry.’ The latter sort of geometry has to do with a symbolic and phantasy consciousness that makes differences in an otherwise uniform sensible plenum. Merleau-Ponty is at odds with the idea of an abstract geometry that constitutes its own field. His phenomenology would have this abstract geometry reduced to pre-geometry, and this pre-geometry further reduced to the sensible, which is then not uniform at all. The problem becomes more precisely how the separations of the sensible and the differences between things are already symbolic.The paper suggests that a way to think through Merleau-Ponty’s problem can be found in a January 1960 Working Note to The Visible and the Invisible called “The Invisible, the negative, vertical Being.” This note, with its references to “joints and members” and “disjunction and dis-membering” is very likely a reference to, and correction of, Heidegger’s 1943 essay on Heraclitus. The point for Merleau-Ponty seems to be a radicalization of the Heraclitean coinstantiation of opposites. His reading allows him to resist understanding the sensible as a self-identity and replace it instead with disarticulation, disintegration, or even perhaps disruption. To show this very disruption is, the paper finally argues, to show the sensible with its phantasmagorical character. Finally, by way of conclusion, the paper takes a turn to sculpture precisely to discuss how we might grasp a sensible that is not uniform but always being shaped and differentiated; and how the symbolic is exactly in, enjambed with, this spacing apart. Ce texte commence par signaler une difficulté évidente dans l’oeuvre tardive de Merleau-Ponty : défaire la séparation décisive entre la connotation linguistique et la dénotation, défaire la séparation décisive entre la signification linguistique et le monde sensible. Cette difficulté requiert de comprendre comment le sensible et le symbolique se trouvent dans une sorte de relation spontanée. Comment cela se fait-il? L’histoire de ce problème remonte à Husserl et à son Origine de la géométrie. Pour Husserl, la « géométrie abstraite » est comprise comme un champ qui se constitue lui-même et qui se détache d’une « pré-géométrie ». Cette dernière sorte de géométrie est liée à une conscience symbolique et imaginative qui fait des différences dans un espace sensible autrement uniforme. Merleau-Ponty n’est pas favorable à l’idée d’une géométrie abstraite qui constitue son propre champ. Sa phénoménologie vise à réduire la géométrie abstraite à la pré-géométrie, et cette dernière au sensible, qui dès lors n’est plus du tout uniforme. Le problème consiste alors à montrer comment les séparations du sensible et les différences entre les choses sont toujours déjà symboliques.La contribution ici suggère qu’une solution peut être trouvée dans la note de travail du Visible et l’invisible de janvier 1960, intitulée « L’Invisible, le négatif, l’Être vertical ». Cette note, qui parle de « jointure et membrure » et de « dis-jonction et dé-membrement », est sans doute une référence à, et une correction l’essai de Heidegger de 1943 sur Héraclite. Le propos de Merleau-Ponty semble être une radicalisation de la co-instantiation héraclitéenne des opposés. Son interprétation lui permet de résister à la conception du sensible comme identité à soi et de la remplacer par la désarticulation, la désintégration ou même peut-être le bouleversement (disruption). Le fait de montrer ce bouleversement consiste précisément à montrer le sensible avec son caractère fantasmatique. Finalement, en guise de conclusion, le texte se tourne vers la sculpture pour discuter une notion du sensible comme étant non uniforme mais toujours déjà formé et différencié, et une notion de symbolique comme étant exactement dans cet espacement. Questo testo inizia portando l’attenzione su un evidente problema nella filosofia dell’ultimo Merleau-Ponty: annullare la separazione decisiva tra connotazione linguistica e ciò che viene denotato, annullare la separazione decisiva tra significato linguistico e mondo sensibile. Questo problema richiede una comprensione della relazione spontanea tra sensibile e simbolico. Come ciò è possibile? La storia di questo problema risale fino ad Husserl, ed in particolare al suo L’origine della geometria. Per Husserl, la ‘geometria astratta’ è da concepire come un campo che si costituisce da sé, distaccandosi da una ‘pre-geometria’. L’ultima tipologia di geometria ha a che fare con una coscienza simbolica e immaginativa, che crea delle differenze in un altrimenti uniforme plenum sensibile. Merleau-Ponty non concorda con l’idea di una geometria astratta che costituisca il proprio campo. La sua fenomenologia intende piuttosto riportare la geometria astratta alla pre-geometria, e quest’ultima al sensibile, che quindi è tutto fuorché uniforme. Il problema diventa allora, più precisamente, come mostrare che le separazioni del sensibile e delle differenze tra le cose siano già simboliche. Questo testo suggerisce che una soluzione per pensare questo problema in Merleau-Ponty possa essere trovata in una nota di lavoro al Visibile e l’invisibile del gennaio 1960, intitolata “L’invisibile, il negativo, l’essere verticale”. Questa nota, con i suoi riferimenti a “spazio topologico e il tempo di congiunzione e di membratura”, e ancora “di dis-giunzione e dis-membramento”, è molto probabilmente un riferimento, ed insieme una correzione, al saggio di Heidegger su Eraclito del 1943. Il punto, per Merleau-Ponty, sembra essere una radicalizzazione del concetto eracliteo di compresenza degli opposti. La sua lettura gli permette di resistere alla comprensione del sensibile come un’auto-identità, per rimpiazzarla invece con una disarticolazione, disintegrazione, o perfino interruzione. Mostrare quest’interruzione è, come argomentiamo in questo testo, mostrare il sensibile con il suo carattere fantasmagorico. Infine, tirando le conclusioni, questo testo si rivolge alla scultura, per discutere come sia possibile afferrare un sensibile che non sia uniforme, ma sempre in-formato e in differenziazione, e come la simbolica sia sempre in – e intrecciata con – questa spaziatura, questa distanziazione. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Beyond the Aesthetic Garden: Politics and Culture on the Margins of Fin-de Siècle ViennaScott SpectorThe rhetorical structure supporting Carl E. Schorske’s seminal Fin-de-Siècle Vienna: Politics and Culture 1 is frankly exposed. The argument—which may have single-handedly changed the discipline of cultural history—is an apparently simple one, and it is reasserted in this series of essays on diverse areas of cultural activity through the use of recurring metaphors. Schorske’s famous (...) thesis (modern culture as political surrogate for a marginalized liberal bourgeoisie) is represented through constantly resurfacing dichotomies: liberal political engagement and (vs.) the aesthetic “garden” of modern art; history and (vs.) desire and “the psyche”; politics and (vs.) culture. All of this is well known. In this essay I want both to assess our necessary debt to Schorske’s powerful work and to suggest what lies beyond it. If at the end of our own century we want to rethink Schorske’s founding project of an interdisciplinary cultural history of Central Europe, we cannot limit this initiative to the obviously important question of extending the contents of cultural inquiry (from high culture to popular culture, from the hegemonic nationality to non-German-speaking cultures, from patriarchy to the voices of women). The formal question packed up in the image of refuge from the historico-political in the aesthetic garden or the fortress of the psyche may offer most productive access to rethinking the Schorskean paradigm, instead of merely expanding its scope. For these reasons I want to focus on the rhetorical structure of Schorske’s essay collection against that of aesthetic and critical works from the Central European turn of the century. 2 Schorske’s historical analysis seems to be at odds [End Page 691] with these projects seen on their own terms—but to utter this is ironically to reiterate the tension between history or politics and aesthetic modernism articulated in the essays.The question at hand is hence two-fold. First of all, what is ideologically at stake in a historiographical construction that identifies Central European modernist aesthetic innovation as an escape from history and politics? Schorske’s interdisciplinary project of contextualization (his significant subtitle “Politics and Culture” has become an unshakable legacy for cultural history) is peculiar insofar as his diagnosis is that culture became increasingly isolated from political life. I have chosen to focus on the rhetorical opposition of “center” and “margin” in this essay because of Schorske’s implicit claim that the very class which was central to nineteenth-century Austrian liberalism—the German-speaking bourgeoisie—was “marginalized” by the rise of hostile anti-liberal ideologies (the “culture-makers” were “alienated along with their class in its extrusion from political power”). 3 As I hope to show, this geometry of center and margins was at the heart of the most innovative and potentially explosive play within so-called aestheticist texts around 1900. This brings us to the second half of this two-fold question: how might the “aesthetic moment” of the Central European fin de siècle be seen if we wish to go beyond Schorske’s diagnosis of a forced retreat into the temple of art, the fortress of the psyche, or the aesthetic garden? To get at both sides of this question—the ideological valence of the aesthetic moment and of its histories—I will focus on literary products, in particular on essays, for several reasons. One of these is that Schorske’s synthetic work is, after all, itself a collection of essays published over two decades.The privileging of the essay genre in coming to terms with the problem of turn-of-the-century aestheticism was established long before Schorske’s work. If we can find early examples of this generic strategy in fin-de-siècle authors such as Musil and Broch, the paradigmatic instance—and the place where the function of the essay-critique within aestheticist criticism is worked through—is the collection of Georg Lukács, Soul and Forms. 4 Several positions are taken in that work which may be useful to an understanding of Schorske’s essay project: Lukács asserts the essay’s fragmentary nature at... (shrink)

In Andrej Belyjs Roman Petersburg (entstanden 1911-1913) kulminiert der Mythos dieser Stadt und wird zugleich ironisiert. Stadt und Roman sind bestimmt von Gegensätzen wie dem von Russland und Europa, Vater und Sohn, Zarismus und Revolution etc. Im Krisenjahr 1905, in dem der Roman spielt, sind diese Gegensätze allenthalben am Werk, aber sie erlauben dennoch keine Orientierung: Denn die Staatsmacht steht nicht nur rebellischen Elementen gegenüber, sondern übt selbst Terror aus, z.B. durch Doppelagenten. In einem Klima gegenseitiger Bespitzelung und Bedrohung sind (...) alle Zuordnungen ungewiss. Welche Funktion haben hier die zahlreichen geometrischen Referenzen: Punkt, Linie, Quadrat, Kubus und die Rede von der ›vierten‹ oder ›n-ten Dimension‹? Der Essay geht diesen Fragen nach und stellt u.a. Beziehungen zur Popularisierung der n-dimensionalen Geometrie und zu avantgardistischer Kunst(-theorie) her, namentlich zu Kandinsky und Malewitsch. In Andrey Bely’s novel »Petersburg« (written between 1911 and 1913), the myth of Petersburg comes to a peak while being simultaneously ironized. The city and the novel are determined by oppositions such as those between Russia and Europe, father and son, tsarism and revolution, and so on. Set in 1905, a year of deep crisis, these oppositions are at work everywhere in the story. Nevertheless, they do not permit any understanding of what is going on: this is because the state is not only challenged by rebellious elements, but also acts as a terrorist itself; for instance, with the help of double agents. In a climate of threat and fear, everything becomes uncertain. What is the function of the numerous geometrical references – point, line, square, cube, and of the ›fourth‹ or the ›nth dimension‹ – in such a context? The essay asks these questions and links the novel to the popularisation of n-dimensional geometry, as well as to the art and art theory of the early 20th century, namely to Kandinsky and Malevic. (shrink)

Dire d’une proposition que, nécessairement, elle est nécessairement vraie, c’est affirmer incomparablement plus que ce que l’on affirme en disant simplement qu’elle est nécessairement vraie. C’est en effet, intuitivement, affirmer qu’elle est nécessaire quelle que puisse être la donnée de tous les mondes possibles à l’aune de laquelle sa nécessité est établie. C’est faire de cette don­née elle-même un possible parmi d’autres, et faire ainsi référence à des mondes possibles d’ordre supérieur. Cet article vise à formaliser la notion de monde (...) possible d’ordre supérieur au moyen d’outils empruntés à la géométrie riemannienne. Le cadre sémantique proposé repose sur une collection ouverte de mondes possibles de niveaux croissants, par opposition à l’héritage leibnizien d’une totalité close de mondes possibles. (shrink)

Dire d’une proposition que, nécessairement, elle est nécessairement vraie, c’est affirmer incomparablement plus que ce que l’on affirme en disant simplement qu’elle est nécessairement vraie. C’est en effet, intuitivement, affirmer qu’elle est nécessaire quelle que puisse être la donnée de tous les mondes possibles à l’aune de laquelle sa nécessité est établie. C’est faire de cette don­née elle-même un possible parmi d’autres, et faire ainsi référence à des mondes possibles d’ordre supérieur. Cet article vise à formaliser la notion de monde (...) possible d’ordre supérieur au moyen d’outils empruntés à la géométrie riemannienne. Le cadre sémantique proposé repose sur une collection ouverte de mondes possibles de niveaux croissants, par opposition à l’héritage leibnizien d’une totalité close de mondes possibles. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 39.3 (2001) 448-449 [Access article in PDF] James Van Cleve. Problems from Kant. New York and Oxford: Oxford University Press, 1999. Pp. xii + 340. Cloth, $45.00. The author acknowledges his debt to the "great Kant books of the 1960s, Jonathan Bennett's Kant's Analytic, and P. F. Strawson's The Bounds of Sense."Their analytical spirit lives on in this book, but the analyses are (...) fresh, take surprising turns, and make unexpected connections to current concerns. In seventeen chapters Van Cleve deals with necessity and the a priori, space and geometry, incongruent counterparts, time, the ideality of matter, experience and objects, substance in the First Analogy, the Second Analogy, noumena, problems of the self, rational theology, and Kant and contemporary irrealism. To sustain the flow of argument, the discussion of certain details, often important, is postponed and taken up in fifteen appendices, and there are nearly seventy pages of careful notes.Chapter 10, "Noumena and Things in Themselves," (134 ff.) criticizes in detail the prevailing "one world" or "double aspect" view of objects, namely that Kant allows just one set of objects, but two ways of "considering" them. Things in themselves are the very same objects as appearances, but considered as they are independent of our experience of them, a view developed in Henry Allison's Kant's Transcendental Idealism. Against this Van Cleve maintains (and I concur) that "appearances are intentionalia, and that intentionalia are constructions out of states of perceivers" (150). The distinction is between two separate universes of discourse, "not between two ways of discoursing [End Page 448] about the same objects" (ibid.). There is no space to report the sharply argued detail that led him to hold the double aspect view to be unfathomable and incoherent. He takes Kant to be a phenomenalist but holding to a version far removed from the reductionist phenomenalism of Russell, Broad, Price, and others. Here the basic elements are not made out to be indubitable givens, but are, rather, sensations. The structures built on them are, according to Van Cleve, "virtual objects" in the sense in which Quine speaks about "virtual classes." To say that something is an element of a class {F:Fx} is only to say that it is F. So, likewise, to say that, e.g., a patch of red exists is short for saying that a certain representation occurs. Carnap's Kantian masterpiece, TheLogical Structure of the World (The Aufbau) could have been consulted as a comprehensive guide to the kind of virtuality here envisaged.It is Van Cleve's stated aim never to discuss Kantian arguments without making their premise and conclusion structure clear (viii). Quite a few of them he finds persuasive; in other cases the arguments are valid, but one or the other premise doubtful. Sometimes he sides with Kant against his critics, sometimes the opposite, and occasionally, as in the case of the Second Analogy, it remains unclear what Kant's argument was. He has shown in this and earlier publications that all reconstructions are flawed, so that either Kant had no good argument, or his argument has not yet been discovered.The title Problems from Kant must be taken seriously. The book rarely deals with textual problems, historical context, or translation difficulties. Rather, the focus is on perennial and still compelling problems. Themes from the First Critique are tracked to and connected with current concerns and sometimes recently ventilated issues are traced back to Kant, as in the last chapter, where the question is raised if Kant was an internal realist in the style of Putnam, or perhaps an anti-realist like Dummett. The contrast with Kant shows that like all complex positions, these are shopping baskets with several goods in them, some in Kant's inventory, others not.Logical notation is used sparingly, and only where it clarifies—no attempt here to shoehorn arguments into unneeded symbolism. Deft examples, Kant's weakness, are a strength of this book. To explain the logic of considering Van Cleve asks, "Consider me apart from my shoes, am I barefoot... (shrink)